The Shade Smooth option is ignored when baking a displacement map
Now it's much more clear and you put your "real" question. 

It seems you wish to bake displacement (ie: geometry) from shading (ie lighting): it will never work imho. 

Use a real "smoothing" option, like more subsurf



so, why "Shade Smooth" doesn't work when baking a displacement map, unlike the "normal map" (that works)?

because "smooth shading" is, in fact, just a shading trick: it alters the normals, to let the object appear smooth (as if it had more geometry, which it has not), so the normal baking takes it into account.

the "displacement baking" instead needs to bake the geometry "height levels" into color levels (the baked image), but since "smooth shading" leaves the geometry untouched, you get the same identical result as without the smooth shading...